Conley theory has proven to be a useful tool for the analysis of global dynamics. Its power arises from the fact that the index provides information about the existence and structure of dynamics, but remains constant as long as isolation is preserved. In particular, it allows one to compute or model the dynamics using one - preferably simple to understand - system and draw conclusions about the dynamics of other systems. The motivation for the work discussed is that there are a variety of problems which arise in study of the dynamics of PDEs for which the natural phase space of the known system is different from the natural phase space of the system of interest. Attempting to apply Conley theory in this setting immediately leads to two problems: (1) one needs to be able to deduce isolation in one phase space from different phase space, and (2) one needs to be able to argue efficiently that the Conley index remains unchanged. I will discuss an approach to dealing with these two problems. This is joint work with G. Raugel.